Finite presentation and purity in categories $\sigma [M]$

Tom 99 / 2004

Mike Prest, Robert Wisbauer Colloquium Mathematicum 99 (2004), 189-202 MSC: Primary 16D90; Secondary 18E15. DOI: 10.4064/cm99-2-4


For any module $M$ over an associative ring $R$, let $ \sigma [M] $ denote the smallest Grothendieck subcategory of ${\rm Mod}\hbox {-}R$ containing $M$. If $ \sigma [M]$ is locally finitely presented the notions of purity and pure injectivity are defined in $ \sigma [M]$. In this paper the relationship between these notions and the corresponding notions defined in ${\rm Mod}\hbox {-}R$ is investigated, and the connection between the resulting Ziegler spectra is discussed. An example is given of an $M$ such that $ \sigma [M]$ does not contain any non-zero finitely presented objects.


  • Mike PrestDepartment of Mathematics
    University of Manchester
    Manchester M13 9PL, UK
  • Robert WisbauerMathematisches Institut der
    D-40225 Düsseldorf, Germany

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